homology and cohomology reduction algorithms and applications. · homology and cohomology reduction...

Homology and cohomology reductionalgorithms and applications.

Paweł DłotkoInstitute of Computer Science, Jagiellonian University,

Kraków, Poland.

What do people think about it?

”Topology! The stratosphere ofhuman thought! In the twenty-fourthcentury it might possibly be of use to

someone...”The First Circle, A. Solzhenitsyn (found by Vin de Silva and Robert

Ghrist)

Topological nature of Nature?

I Phenomena in many theories (far from topology) has atopological ( co-homological ) nature:

I Maxwell equations in electromagnetism,I Coverage problem and hole detection in sensor network theory,I Pattern and image recognition,I Phase transition in physic,I Hard ”regions” of NPC-problems in computability theory,I ...

Introduction

How homology is computed inCracow – the reduction

algorithms.

How homology is computed in Cracow – the reductionalgorithms.

I Pure mathematics – to compute (co) homology Smith NormalForm alg. suffices – requires only elementary matrixoperations – easy and well established algorithm.

I Hypercubical complexity.I In low dimensions 2 (Betti numbers + gens.) and 3 (Bettinumbers when the input is suitable – Delfinado-Edelsbrunner,1995) can be computed much faster.

I Higher up one can rely only on slow SNF computations.


Question: How toovercome the problemwith subpercubicalcomplexity?


Answer: Make its input assmall as possible!


Idea! The same homology, different number of cells:


Elementary reductions by J. H. C. Whitehead:


The main reduction algorithms used by us:

1. Elementary reductions (by J.H.C Whitehead),

2. Coreductions (by B. Batko and M. Mrozek),

3. Acyclic subspace method (by M. Mrozek, P. Pilarczyk, N.Żelazna),

The Coreduction homology algorithm.

Summary of elementary reductions & coreductions.

I In many cases elementary reductions + coreductions reducethe complex to the S-complex with zero boundary map.

I The reduction algorithms provide both Betti numbers and thehomology generators.

I One can prove that in R2 do all the job.I Numerical experiments show that in higher dimensions theyalso reduce a lot.

Acyclic subspace algorithm.

I Elementary observations:

Hn(X ) =

Hn(X ,A) n > 0 ,

Z⊕ Hn(X ,A) n = 0 ,

when A is acyclic.


What’s left after acc. subspace being removed:


I No theorems saying how big percentage of the cells can beremoved,

I In practical applications the reductions goes very deep.

Numerical experiments.

Size bbm CR AS

28800 7.99 0.14 0.1351200 15.2 0.23 0.20115200 36.7 0.55 0.39204800 65.0 0.97 0.63320000 104. 1.56 0.96Torus (dim=2, emb=3)

Size bbm CR AS

74341 16.7 0.44 0.27132321 36.8 0.80 0.39206901 71.5 1.27 0.61298081 105. 1.80 0.84405861 163. 2.52 1.23Bing’s House (dim=2, emb=3)

Enviroment: 3.6 GHz Pentium 4, 2GB memory, Windows XPI bbm - building boundary matrices

Numerical experiments.

Size bbm CR AS

12522 14. 0.26 1.722280 36. 0.51 3.034830 86. 0.90 4.850172 184. 1.4 7.068306 326. 2.1 10Klein bottle (dim=2, emb=4)

Size bbm CR AS

937 0.56 0.01 0.303745 3.1 0.07 1.98425 8.6 0.15 4.114977 18. 0.30 5.423401 39. 0.48 8.7Projective plane (dim=2, emb=4)

Enviroment: 2.2 GHz Pentium 4, 1GB memory, LinuxI bbm - building boundary matrices

The representation problem.

I Usually the data for computing given a priori.I But sometimes, we are generating it by ourself,I Some representations are more suitable for one kind ofapplications, another for others,

I for instance- not worth to triangulate the data obtained fromsome images (cubes are more feasible)

The representation problem.

I Selecting a good representation often results in decreasing thenumber of cells by a few orders of magnitude.

Simplicial versus Cech representation

The bibliography.

CAPD, http : //capd .ii .uj .edu.pl .

Chomp, http : //chomp.rutgers.edu.

T. Kaczynski, K. Mischaikow, M. Mrozek,Computational Homology, Springer-Verlag, New York, 2004.

B. Batko, M. Mrozek Coreduction homology algorithm,Discrete and Computational Geometry, 41(2009), 96-118.

M. Mrozek, P. Pilarczyk, N. Żelazna, HomologyAlgorithm Based on Acyclic Subspace, Computers andMathematics, 55(2008), 2395–2412.

M. Mrozek, Cech Type Approach to Computing Homologyof Maps, to appear in Discrete and Computational Geometry.

Applications of computational (co)homology.

I Part 1: Homology of nodal sets using homologycomputations for regular CW-complexes.

I Part 2: The coverage problem in the sensor network.

Applications of computational (co)homology.

I Part 3:Where Poincare meets Maxwell, Discrete Geometricapproach to Maxwell’s equations.

Homology groups of regular CW-complexes –computations of homology of nodal domains.

Part 1

(joint work with Tomasz Kaczynski, Marian Mrozek and ThomasWanner.)

Homology groups of nodal domains.

I High importance of nodal domains in applied science.I For instance, for the functionu : R× R 3 (t, x) → u(t, x) ∈ R the nodal domains areN+(t) = x : u(t, x) ≥ 0 and N−(t) = x : u(t, x) ≤ 0.

I An example of the nodal domain of a trigonometricpolynomial u presented in the previous slide.

I Studying the evolution of Betti numbers of nodal domainshelps to understand the dynamical system induced by u(papers by K. Mishaikow, T. Wanner, S. Day, W. Kalies).

Homology groups of nodal domains.

I Among the reasons which make the topic interesting are the,so called, phase separation phenomena.

I Suppose some object consists of two materials the atoms ofwhich are well mixed.

I Roughly, when the temperature of the object drops downrapidly from some level, the two materials separate.

I This results in local clustering of one material inside another.I In consequence, the mechanical properties of the object getworse

I Such a situation typically happens in pipes guiding steam inpower stations (their lifetime is shorter then excepted).

Approach used so far.

I Lots of subdivisions have to be done close to the boundary ofthe nodal domain.

I So far, after the nodal domain was fixed, the subdivisions wasdone to obtain cubical complex with the cubes having equalsizes...

I ... so the size of the complex grows drastically

Approach used so far.

I Even if one is able to remove most of the cubes withelementary reductions, the whole process is still very costly.

I Idea! Compute the homology of the non uniform size cubicalgrid:

How to compute homology of non uniform size cubicalgrid?

I CW- complex.I The existing software designed for cubical, simplicialhomology.

I No code available for general CW-complexes.I To maintain the computations (SNF) the incidence numbersof cells are essential.

I In the standard theory the incidences are given either byBrouwer degree of some continuous function or by somerelative homology of cells (triangulation of cells required!)

I Both approaches hardly feasible.I Do we face the wall then?

Regular CW-complexes. Trade between computability andgenerality.

I Subclass of all CW-complexes for which computation of theincidence is easy exists (regular CW-complexes).

I This subclass was introduced by William Massey in ”Basiccourse in Algebraic Topology”.

I An n-cell en is attached to the closed subset K of theHausdorff space if there exists a continuous mapf : Bn1 (0) → en, called the characteristic map, such that fmaps the open ball Bn1 (0) homeomorphically onto e

n andf (∂Bn1 (0)) ⊂ K .

I A CW-complex is said to be regular if for each cell en, wheren = 1, . . . ,N, there exists a characteristic map f : Bn1 (0) → enwhich is a homeomorphism.


I For the regular CW-complexes the incidence numbers havethe following properties:

(a) If en−1µ is not a face of enλ, then αnλµ = 0.

(b) If en−1µ is a face of enλ, then αnλµ = ±1.

(c) If e0µ and e0ν are the two vertices of the 1-cell e

1λ, then

α1λµ + α1λν = 0.

(d) Let enλ and en−2ρ be two cells of K such that en−2ρ is a face

of enλ. Furthermore, let en−1µ and en−1ν denote the unique

(n − 1)-cells en−1 such that en−2ρ ⊂ en−1 ⊂ enλ. Then theidentity αnλµα

n−1µρ + αnλνα

n−1νρ = 0 holds.

I Using these properties it is possible to create a linear timealgorithm which computes the incidences.


I The family of regular CW-complexes is wide enough to coverall, known to us, practical applications.

I The nonuniform size cubical complexes are regularCW-complexes.

I Also in the mathematical modeling in electrical engineeringthe octahedron meshes are very popular.

I Soon the CAPD (Computer Assisted Proofs in Dynamics,capd .ii .uj .edu.pl) software will provide the functionality ofcomputing the (co)homology groups of arbitrary regularCW-complexes.


I So far the code was implemented and tested for 2-dimensionalnodal domains.

I In this case no algebraic computations are necessary to gethomology, elementary reductions + coreductions do all thejob.

I When compared with the coreduction algorithm based onregular size grid, a speed up of the order of magnitude rangingfrom two to three was observed.

I The CW complex is also highly efficient with respect tomemory usage. Now we are able to get homology of exampleswhich were far beyond the reach of the previous method dueto lack of memory.

Speed up with respect to previously fastest algorithm andthe average time of the algorithm.

The bibliography.

P. Dłoko, T. Kaczynski, M. Mrozek, T. Wanner,Coreduction Homology Algorithm for Regular CW-Complexes,in preparation.

K. Mischaikow, T. Wanner, Probabilistic validation ofhomology computations for nodal domains, Annals of AppliedProbability, 2007.

S. Day, W. D. Kalies, K. Mischaikow, T. Wanner,Probabilistic and numerical validation of homologycomputations for nodal domains, Electronic ResearchAnnouncements of the AMS, 2007.

P.D, T. Wanner Code for computting homology of planarnodal domains, online soon...



The coverage problem in the sensor network.

Part 2,Homology computations on the sensor network,

(joint work with Robert Ghrist, Mateusz Juda and Marian Mrozek.)

Formulation of the problem.I The importance of the coverage problems in the sensornetworks (security, mine field sweeping, ad hoccommunication networks).

I Systems with stationary nodes in R2 with radially symmetriccoverage domains.

I Verify if the given set D such that ∂D = F is covered by thedisks of a given radius centered in the sensors.

How the problem was solved so far?

I Computational geometry approach (Delaunay or Voronoidiagrams).

I Precise geometry of the domain,I known localization of the nodes,I then we have exact solution.

I Probabilistic methods.I Big enough number of the sensors,I uniformly distributed on the regionI implies coverage of the region with high probability (but noguarantee).

I What we wish to have?I Deterministic Yes/No or Yes/No/Do Not Know algorithmI arbitrary geometry of the region,I no assumptions about the placement of the sensors.

The Rips complexes

I Problem of computing the homology groups of a union of setsis classical (nerve of the cover).

I Let X1, ...Xn be the points where sensors are placed.I Let rcom be the radius of communication of a sensor.I The Rips complex associated with the set X = X1, ...Xn is asimplicial complex whose simplices are subsets V ⊂ X suchthat for each X ,Y ∈ V , d(X ,Y ) ≤ rcom

The radius of a cover.

I Each sensor has some radius of coverage rcov .

I Putting rcov ≥√13 rcom ≈ 0.577rcom enables us to ”fill all the

gaps”.

Example of Rips complex (see V. Silva, R. Ghrist [3])

The homological criterion

I The coverage criterion based on the relative homology of theRips complexes.

Theorem (Silva, Ghrist, 2006)If there exists a homology class [α] ∈ H2(Rrcom ,Frcom) such that∂α 6= 0, then the considered region is covered by the balls of aradius rcov centered in the given sensors.

I To use this (partial) criterion we need to compute the secondrelative homology generator...

I check its boundary...I When this criterion holds, some sensors can be turned off(power-saving mode).

Rips complexes in the coverage problem

I Rrcom denotes the Rips complex based on the set ofdistributed sensors and radius rcom.

I The fence complex Frcom is a subcomplex of Rrcom .I A geometric realization of the fence complex is assumed to bea Jordan curve in the plane.

I The Silva-Ghrist criterion is used to check the coverage of theinterior of this curve.

What we should do to check this criterion

I The sensors need to be distributed in the given set.I The fence sensors have to know that they belong to the fence.I Each sensor has to get to know all its neighbors (it suffices toconstruct the Rips complex).

I Construct the Rips complex.I Compute relative homology of the complex.I Last two things can be done in theory after gathering all theinformation from all the sensors into one computer...

I ...which is not easy and not effective for a real–sizedcomplexes.

Idea

So maybe we can try to make these computations ina distributed way on the sensors themselves?

I Doing algebraic homology computations in the distributed wayis not easy.

I Why not to use the reduction procedures, that can beimplemented in a purely combinatorial way?

I In some cases the (co)reductions can reduce the complex upto the homology generators.

What kind of sensors do we need to maintain the parallelcomputations?

I Each sensor is identified by unique integer label and equippedwith a processor and radio transmitter.

I Each sensor can read the labels of its neighbors lying notfarther than rcom (the communication range).

I The sensor can communicate with the sensors lying notfarther then rcom.

I The sensor has the radius of detection rcov such that

rcov ≥√13 rcom.

I The sensor does not have any metric information (neithercoordinates, nor distance, nor direction to other sensorsknown).

What kind of sensors do we need to maintain the parallelcomputations?

I Each sensor can keep the list of simplices.I Each simplex S from the list can keep the followinginformation:

I Labels of all the sensors that belong to S ,I labels of all the sensors that do not belong to S and are in thecommunication range with all sensors that belong to S,

I lists of faces and cofaces of the simplex.

I For the efficiency and simplicity we need to know exactly inwhich sensors each simplex from the Rips complex should bekept.

How the Rips complex can be effectively (withoutredundancy) represented in the sensor network?

I Each sensor has a unique integer label.I Let [X1, ...Xn] be the representation of the simplex S in theRips complex with sensor’s labels.

I Convention – if the dimension of a simplex S is greater orequal 1, then the information about S is kept in the twosensors having the lowest labels among [X1, ...Xn].

I 0-dimensional simplices are kept only in one sensor.I Convention – all the decisions about the simplex S are takenby the sensor with the smallest label in S .

I So for each simplex S all sensors that ”rule” the boundary ofS have the information about S .

Construction of the Rips complex by the sensors alone

I Once the sensors are distributed, they need to communicateto get to know the labels of all its neighbors.

I Moreover every sensor needs to know the labels of theneighbors of its neighbors.

I No more communication is needed to construct the Ripscomplex.

I The communication graph is a 1-skeleton of the Rips complexwe are looking for.

I From the presented convention each 1-simplex is kept in theboth of its endpoints.

Construction of the Rips complex by the sensors...I Each simplex S keeps the list of the sensors X1, ...Xn thatare neighbors of each sensor involved in S .

I S ∪ Xi for i ∈ 1, ...n is a simplex in a Rips complex we arecreating.

I We have to create the simplex S ∪ Xi in the right sensors tofollow the given convention.

I The simplex is created in a sensor if the sensor is in charge ofS in the following two situations:

I The label of Xi is greater than any label of the sensor in S .I The label of Xi is smaller than any label of the sensor in S

I In this way all the simplices that belongs to the Rips complexwill be created,

I each simplex will be created by at most two sensors, and onlyone of them will be in charge of the simplex.

I The presented procedure can be easily implemented on theparallel architecture like the sensor network.

Example.

Fence reduction procedure

I Silva-Ghrist’s criterion needs relative homology groupgenerators [α] ∈ H2(Rrcom ,Frcom).

I Computation of relative homology is needed.I The fence complex Frcom is a closed subcomplex of thecomplex Rrcom .

I In this case H2(Rrcom ,Frcom) is isomorphic to H2(Rrcom\Frcom).I So we simply remove the 0 and 1 dimensional simplices thatbelong to the fence from the constructed Rips complex.

I This can be done by the sensors themselves if each sensorknows if it belongs to the fence or not (we assume they knowthat).

(Co)reduction algorithms

I The created Rips complex is usually a really big structure.I The simplices of arbitrary dimension can appear in it.I This is not a good input for the homology computationalgorithm, we need to reduce the input as far as possible.

I To do so the combination of the elementary reduction andcoreduction algorithm will be used.

(Co)reduction algorithms

I Elementary reduction – the retraction procedure:

I The (co)reduction algorithm is a kind of deal between asimplex and its (co)boundary.

I But from the construction of the Rips complex we know, thatthe sensors that have the information about a simplex S alsohave the information about its boundary.

I Once the technicalities are fixed the sequence of elementaryand coreductions works perfectly.

How far can we reduce the complex? Do we need algebraiccomputations?

ConjectureFor the relative homology of a Rips complex built on the basis ofthe planar point set the sequence of elementary and co –reductions reduce the complex up to its homology generators.

I By the generator in the reduced complex we mean the simplexwithout boundary elements.

I The idea behind this – even if the dimensions of simplexes inthe Rips complex are high, the topology of it almost ”planar”(in planar case the reductions can do all the job) ...

I this is what we see from the numerical experiments.I Even if the Conjecture is not true, the complex that remainsafter (co)reductions is very small...

I and can be gathered by one of the sensors and the homologycan be computed.

Can this work? Can we prove the algorithm is correct?

I In the papers dealing with elementary and co – reductionsonly proofs of the correctness of the sequential algorithms areprovided.

I We are using a parallel version of these algorithms.I The parallel execution of the presented algorithm is equivalentto certain execution of the sequential algorithm.

I We demonstrate that the maps induced in homology by theparallel execution are isomorphic to the maps induced by thesequential algorithm...

I ... so both parallel (co)reductions and pulling back thegenerator are correct algorithms.

The bibliography for this section.P. Dłotko, R. Ghrist, M. Juda, M. Mrozek,Distributed computation of homological coverage in sensornetworks, in preparation.

V. de Silva, R. Ghrist, A. Muhammad, Blind Swarmsfor Coverage in 2-D, Proc. Robotics Systems & Science, 2005.

V. de Silva and R. Ghrist, Coordinate-free coverage insensor networks with controlled boundaries via homology, Intl.J. Robotics Research, 2006.

B. Batko, M. Mrozek, Coreduction Homology Algorithm,Discrete and Computational Geometry, 2009.

Zin Arai, Kazunori Hayashi, Yasuaki Hiraoka,Mayer-Vietoris sequences and coverage problems in sensornetworks, preprint.



Where Maxwell meets Poincare.

Part 3,Computational homology in electromagnetism, where Poincare

meets Maxwell.

(joint work with Ruben Specogna and Francesco Trevisan.)

Modelling of real-world objects

I Tetrahedral mesh used in modelling real-world objects.I Discretization of the Maxwell’s equations used, computedvalues constant for each simplex in the mesh.

I Maxwell’s equations can be formulated by using only(co)chains and (co)boundary operator [2], [5].

I In such approach from very beginning theory works on thediscrete level. No approximation of continuous mathematicaloperators needed.

I T − Ω formulation uses this approach. However it requiressome prior knowledge about the topology of the mesh.

What do electrical engineers need the (co)homology for?

I Two regions present in the meshes:I Conducting region, Dc .I Non-conducting (air) region, Da.

I Assumption: Da ∪ Dc is acyclic.I Extra topological information needed to enforce the Ampere’slaw in Da.

I Ampere’s law in the classical physics– integrated magneticfield around a closed loop is related to the electric currentpassing through any surface the boundary of which is the loop.

I Discrete level:I Cycles in Da instead of the closed loops.I Chains having the cycle as a boundary instead of the surfaces.I The magnetomotive force (this is what we are looking for) d isa cochain.

I The integral of a magnetic field over some chain c is theevaluation < d , c >.

Non-trivial topology appears

I If Da – topologically trivial, then each cycle is a boundary.Electric current passing through the boundary is zero (thecurrent can be driven only by elements of Dc).

I It is getting more interesting when Da is homologicallynon-trivial (nonzero β1).

I If c1 in Da is not a boundary in Da, then each chain S inDa ∪ Dc such that ∂S = c1 needs to ”intersect” Dc .

I The information about cycles that are not boundaries in Da isneaded. The basis of such cycles is precisely the basis of thehomology group H1(Da).

Topologically nontrivial Da

What is the topological information we need, though?How the magnetomotive force should be defined?

I Standard approach (used so far in the electrical engineering) –cochain called thick cut has to be found to make the T − Ωformulation computations in the homologically nontrivialregions possible.

I For example in case of simple torus the thick cut d is definedin such a way that for each cycle c that ”goes n times aroundthe branch of a torus” we have | < d , c > | = n.

GSTT, the standard algorithm used by the engineers toconstruct thick cut

I To obtain thick cut it is enough to enforce the evaluation ofthe cochain on all the representatives of homology generators,and enforce zero evaluation on all boundaries (once this isdone, all other cycles are spanned ”automatically”).

I There exists a standard algorithm to compute the thick cut(representatives of homology generators needed at the input).

I So called belted tree is constructed at the beginning by thealgorithm.

I h1, ...hβ1(Da) – representatives of the H1(Da) basis.I A belted tree T in Da – a maximal spanning tree T ′ with theset of edges Eiβ1(Da)

i=1 such that only cycle in T ′ ∪ Ei is asupport of hi .

Generalized spanning tree technique (GSTT)

I When a belted tree T = T ′ ∪ Eiβ1(Da)i=1 is given the following

algorithm is called for each i ∈ 1, ...β1(Da).I For every edge E ∈ T ′ put < ci ,E >= 0 and < ci ,Ej >= δijfor j ∈ 1, ...β1(Da).

I Let L be the list of all 2-simplexes in Da.I while L is not empty

I Take T ∈ LI if the cochain ci is defined for all edges in ∂T , then if< ci , ∂T >6= 0, return FAILURE, else L := L\T .

I if the cochain ci is not defined for a single edge E in theboundary of T , then define the value of < ci ,E > to have< ci , ∂T >= 0 and L := L\T .

How does the GSTT work?

The belted tree

The initial value of a cochain

Which 2-simplexes can be removed form the list L in thefirst step of the algorithm?

Which 2-simplexes can be removed form the list L in thesecond step of the algorithm?

Output

Output...

Combining of the GSTT algorithm with the computationalhomology.

I The basis of H1(Da) is needed to run the presented algorithm.I So far – computed by pure Smith Normal Form or chosen ”byhand” (only toy examples could have been computted).

I GSTT algorithm using the homology generators provided bythe CAPD software was implemented by P.D., RubenSpecogna and Francesco Trevisian in [2].

I Fast reduction methods from CAPD open the field ofcomputations for far bigger meshes than it was possible before.

I However the algorithm leaves some open questions.I Does the algorithm always converge?I Can it return FAILURE for some input?I What is the output from the mathematical point?

Generalized spanning tree technique (GSTT)...

I Let’s try to answer the question ”What is the output of thealgorithm?”

I We assume that the while loop has terminated and noFAILURE was returned by the algorithm.

A few simple lemmas

1. If Da ⊂ R3 is compact and locally contractible then Hi (Da,Z)is 0 for i ≥ 3 and torsion–free for i = 0, i = 1 and i = 2.

2. In a simplicial complex Da, the evaluation of 1−cocyclec ∈ Z 1(Da) on a trivial 1−cycle d ∈ B1(Da) is zero.

3. A 1−cochain c is a 1−cocycle if and only if for every2−simplex S , < c , ∂S >= 0.

4. Conclusion– to define a 1-cocycle c , it is enough to imposethe zero evaluation of c in the boundary of all 2-simplexes.This is exactly what takes place in the GSTT algorithm. Sothe GSTT algorithm returns a cochain.

A few more simple lemmas...

I For two 1−cycles d1 and d2, which differ by a boundary, and a1−cocycle c we have < c , d1 >=< c , d2 >.

I Once the evaluation of a 1−cocycle c ∈ C 1(Da) is known overcycles hiβ1(Da)

i=1 representing the basis of H1(Da), the valueof the evaluation of the cocycle c over any 1−cycle isdetermined.

I In the GSTT algorithm the evaluation of a cocycle is fixedover the H1(Da) basis. As a result it is fixed on all the cyclesin the complex.

The Universal Coefficient Theorem for cohomology.

Theorem (See [1], Th. 3.2)If a simplicial complex Da has (integer) homology groups Hn(Da),then the cohomology groups Hn(Da;G ) are determined by splitexact sequences

0→ Ext(Hn−1(Da),G ) → Hn(Da,G )h−→ Hom(Hn(Da),G ) → 0.

I Theorem will be used for n = 1 and G = Z.I I do not want to define the Ext functor, it is enough to know,that Ext(Hn−1(Da),Z) = 0 if Hn−1(Da) is a free group. It isso in case of n = 1.

I So from the exactness of the sequence we have theisomorphisms: h : H1(Da,Z) → Hom(H1(Da),Z).

How is the map h defined?

I h : H1(Da,Z) → Hom(H1(Da),Z).I For a class [ψ] ∈ H1(Da,Z) we have0 =< δψ, z >=< ψ, ∂z >, since ψ is a cocycle. Consequentlyψ|B1(Da) = 0.

I Let us define the restriction ψ0 = ψ|Z1(Da). Sinceψ0|B1(Da) = 0, ψ0 ∈ Hom(H1(Da),Z). ψ0 is well defined, sincethe value of the ψ is the same for all representatives of thesame H1(Da) class.

I We define h([ψ]) = ψ0 ∈ Hom(H1(Da),Z).

What is the output after all?

The output of the GSTT algorithm is a the basis of H1(Da) group.

I hiβ1(Da)i=1 – representation of H1(Da) basis (the same as the

one at the input of the algorithm).I Take φi ∈ Hom(H1(Da),Z) such that φi ([hj ]) = δij .

I Since H1(Da) – free group, φiβ1(Da)i=1 is a basis of

Hom(H1(Da),Z).

I Since h is an isomorphism, h−1(φi )β1(Da)i=1 forms the basis of

H1(Da,Z).

I The cocycles ciβ1(Da)i=1 obtained by the GSTT algorithm are

in the same cohomology class as the cocycles h−1(φi )β1(Da)i=1 .

I Consequently ciβ1(Da)i=1 forms a basis of H1(Da,Z).

The current work.

I The output of GSTT is the H1(Da,Z) basis.I In some cases GSTT return FAILURE,I its performance is in fact equivalent to the performance ofcoreduction algorithm – but of course we know, that ingeneral we cannot reduce everything,

Possibly applications of what we are doing:

I Rapid and cheap design of practical devices together withtheir optimization,

I Non-destructive testing (IP),I induction heating,I proximity sensors (IP),I metal detectors (IP),I hyperthermia cancer treatment (IP),I ITER! (so far impossible to get the mesh).

The bibliography.

A. Hatcher, Algebraic topology.

P. D., R. Specogna, F. Trevisan, Automatic generationof cuts on large-sized meshes for T-Ω geometric eddy-currentformulation, CMAME 2009.

P. D, R. Specogna, F. Trevisan, Voltage and currentsources for massive conductors suitable with the A-χGeometric Formulation IEEE Trans. Mag. 2010.

P. D, R. Specogna, Critical analisys of spanning treetechniques, submitted.

E. Tonti, On the formal structure of physical theories,monograph of the Italian National Research Council, 1975.



The end.

Thank you for your attention!

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